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st: Rescale using covariance matrix for weighted PCA?


From   "J Jones" <statafan@googlemail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: Rescale using covariance matrix for weighted PCA?
Date   Wed, 21 May 2008 22:25:22 +0100

Hello--I would like to do weighted principal-components analysis
(weighted PCA / WPCA; initialisms added for accessibility in future
searches); however I am not sure how to do so.  I have variables,
which may be organized into sets if need be.  I know how to get
covariance matrices from factor analysis or PCA and I know how to do
simple rescaling such as dividing an original score by a variable's
standard deviation.

Is it possible to rescale a variable by manipulating its covariance
matrix?  References say to weight variables from a particular set by
dividing by the-reciprocal-of-the square-roo-
of-the-eigenvalue-of-the-set's-first-principal-component (W)...  and
also dividing by the standard deviation of the variable.

The exact formula they give is:
Rescale score=(orig score)/(st dev*W).

So the general method involves doing PCA on sets of variables and
using the weighitng factor in order to confer proper weight to each
set.  THen you do facotr analysis on the rescaled scores.
Can somebody help please?  Thank you.
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